# [seqfan] Re: Bi-digital multiplications (concatenated)

Eric Angelini Eric.Angelini at kntv.be
Mon Nov 24 09:22:34 CET 2014

```Hello Frank,

> multiply, not multiplicate

... yes, indeed, my apologizes

> With only one digit, there are no pairs, so the concatenation of the
products is empty

... well, I've done like here:

http://oeis.org/A034049, "multiplicative digital root value 2"

... where the seq starts with 2.

Best,
É.

Le 24 nov. 2014 à 01:30, "Frank Adams-Watters" <franktaw at netscape.net<mailto:franktaw at netscape.net>> a écrit :

1, 2, ..., 9 are not fixed points. They all immediately lead to 0,
which is a fixed point.

(With only one digit, there are no pairs, so the concatenation of the
products is empty. The empty string is another representation for zero.)

P.S. It's multiply, not multiplicate.

-----Original Message-----
From: Eric Angelini <Eric.Angelini at kntv.be<mailto:Eric.Angelini at kntv.be>>
To: Eric Angelini <eric.angelini at skynet.be<mailto:eric.angelini at skynet.be>>; Sequence Discussion list
<seqfan at list.seqfan.eu<mailto:seqfan at list.seqfan.eu>>
Sent: Sun, Nov 23, 2014 5:12 pm
Subject: [seqfan] Bi-digital multiplications (concatenated)

Hello SeqFans,
Here is how we "bi-digital" multiplicate;
1) we always read a number N (like 1235) from left to right
2) we successively multiplicate the pairs of digits we encounter in N
3) we concatenate the results:
1235-->2615
(2615 is the concatenation of 1.2, 2.3 and 3.5 that is 2, 6 and 15).

We now iterate to see what happens:

2615--> 1265--> 21230--> 2260--> 4120--> etc.

I guess that the iterating process has three outcomes:
1) fixed point 0, or 1, or 2, or... or 8, or 9
2) infinite expansion (with or without visible patterns)
3) loop

I've found (1) and (2) but not (3)...

Examples:
184-->832-->246-->824-->168-->648-->2432-->8126-->8212-->1622-->6124-->62
8-->1216-->226-->412-->42-->8
END

185-->840-->320-->60-->0 END

186-->848-->3232-->666-->3636-->181818-->88888-->64646464-->2424242424242
4-->8888888888888-->
INF

If this is of interest, one could submit at least a dozen or so
sequences to the
OEIS:

a) integers ending on 0
A=0,10,20,25,30,40,45,50,52,54,55,56,58,59,60,...
[this is not http://oeis.org/A034048 as
239-->627-->1214-->224-->48-->32-->6
here but -->0 in the OEIS seq "multiplicative digital root value 0"]
b) integers ending on 1
c) integers ending on 2
C=2,12,21,26,34,37,43,62,
... [this is not http://oeis.org/A034049, "multiplicative digital root
value 2"]
j) integers ending on 9
J=9,19,33,91,119,133,191,... [this is not http://oeis.org/A034056]
k) integers expanding for ever
K=186, ?, ?, ?,...
[266 is a member of K, though no immediately visible pattern arises]
l) integers ending in a loop
[are there any?]
m) integers with no predecessor
[like 281]
n) integers with exactly one predecessor [23, for instance: 23<--213]
o) integers with exactly two predecessors [189<--291 or 189<--633]
etc.

P.-S.
We consider that 257,for instance, ends on the fixed point 0 [although
all
intermediary integers "do not exist" (because of leading zeroes
somewhere in the
iteration process: 257-->1035-->0015-->005-->00-->0)]
Best,
É.

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